# A categorical semantics for causal structure

## TL;DR

This paper introduces a categorical framework for modeling diverse causal structures in classical and quantum theories, enabling analysis of processes with fixed or indefinite causal orderings.

## Contribution

It provides a novel categorical construction that encodes fine-grained causal relationships and unifies various types of causal processes, including indefinite causal orderings.

## Key findings

- Framework encompasses classical, quantum, and indefinite causal processes.
- Defines families of processes consistent with arbitrary acyclic causal orderings.
- Derives operational behavior of complex causal structures using diagrammatic axioms.

## Abstract

We present a categorical construction for modelling causal structures within a general class of process theories that include the theory of classical probabilistic processes as well as quantum theory. Unlike prior constructions within categorical quantum mechanics, the objects of this theory encode fine-grained causal relationships between subsystems and give a new method for expressing and deriving consequences for a broad class of causal structures. We show that this framework enables one to define families of processes which are consistent with arbitrary acyclic causal orderings. In particular, one can define one-way signalling (a.k.a. semi-causal) processes, non-signalling processes, and quantum $n$-combs. Furthermore, our framework is general enough to accommodate recently-proposed generalisations of classical and quantum theory where processes only need to have a fixed causal ordering locally, but globally allow indefinite causal ordering. To illustrate this point, we show that certain processes of this kind, such as the quantum switch, the process matrices of Oreshkov, Costa, and Brukner, and a classical three-party example due to Baumeler, Feix, and Wolf are all instances of a certain family of processes we refer to as $\textrm{SOC}_n$ in the appropriate category of higher-order causal processes. After defining these families of causal structures within our framework, we give derivations of their operational behaviour using simple, diagrammatic axioms.